The Ontology of Physics (updated)

The question at issue here is, What sort of entities are fundamental in physics?  E.g., in the late nineteenth century through early twentieth centuries,

whether atoms were real objects or only mnemonic devices for coding chemical regularities.

-- Abraham Pais, Subtle is the Lord (1982), p. 80

The idea of atoms in some sense  goes back to Ancient Greece; and today, they are taken for granted.  So it is surprising to laymen, how long opposition to a Realist take on atoms lasted among some philosophers and physicists (e.g. Ernst Mach).  An opposing view, from a leading German chemist:

Ostwald’s ‘Energetik’, according to which molecules and atoms are but mathematical fictions, and energy, in its many forms, the prime physical reality.

-- ibid, p. 83

Good bluff Rutherford, by contrast, had sucked atoms with his mother’s milk, and claimed that he “could see the little buggers as plainly as a spoon”.

Oddly, the debate on this seemingly practical laboratory matter,  had elements more characteristic of political or theological controversy, in which neither side has a prayer of of convincing the other by rational argument:

The most remarkable fact about the nineteenth century debates on atoms and molecules   is the large extent to which chemists and physicists spoke at cross purposes, when they did not actually ignore each other.

-- ibid, p. 80

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Since that time, a host of new physical building-blocks have been proposed, and in some cases observed: from the unexpected and rather unwelcome muon (I.I. Rabi: “Who ordered that?”), to the massless chargeless neutrino (rather like Bishop Berkeley’s “ghosts of departed quantities”), through quarks, gluons, gravitons & gravitinos, selectrons & electrinos, axions, preons, virtual photons, phonons … such as might make even Rutherford gag.  But several of these are now widely accepted -- not as mere bookkeeping mechanisms, but as entities with further properties to be discovered;  so that the smart money has been on Realism, so far.

Ernest Rutherford,  swallowing an atom  but straining at a quark

Footnote:

We shouldn’t be too hard on old Ostwald  for backing the wrong horse in the Atoms Affair.   Dissident voices today would declare  as epiphenomenal not only atoms, but even the elementary particles of which those are admittedly merely bundles:  demoted to excitation-states of superstrings;  or emerging from the combinatorial-automaton structure of the world. 

Additionally, his Energetik has enjoyed a bit of a revival in some quarters:

  https://de.wikipedia.org/wiki/Energetik_(Philosophie)

More recently, information (or, solemnly, “The Information” -- Wheeler's "It from Bit") has emerged among some as a skeleton-key to everything else in the physical world.

Thus, at the bottom of everything, behind and beyond the Maya of the particle zoo, lies:

Thales:  Water.
Oswald:  Energy.
John Wheeler:  Information.

Stewardess:  Coffee tea or mi-ilk?
The Milesian:   Just water, thanks.

Or, a more recherché candidate, from philosopher Hilary Putnam:

Nothing has more physical significance than spectral measure.

(That might strike the layman as rather a … spectral candidate for the role of firmest substrate of all.)

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The word “ontology” is not one you are likely to overhear at the bus-stop;  nor indeed does a physics-major typically run across it.   But the more we see physics as science (Wissenschaft) rather than a special kind of engineering (the “shut up and calculate” ethos of the years around WWII), the more we meet questions traditionally treated under that rubric -- and indeed, contemporaneously, even under that very name. 

As:

My own position is that the issue of ontology is crucial to quantum mechanics.
-- Roger Penrose,  The Road to Reality (2004), p. 785

Let us now return to the ontology of the consistent-histories approach.  The theory operates with entities called coarse-grained histories.  … The ontological status of the insertion of such a projector set  is still not fully clear. …  A history from a maximally refined set seems to me to provide a strong candidate for what might be regarded as ontologially ‘real’.

-- Penrose,  The Road to Reality (2004), p. 788

Present-day quantum mechanics has no credible ontology … The importance of having an ontologically coherent quantum mechanics cannot be over-estimated.
-- Penrose,  The Road to Reality (2004), p. 860, 865

Re the notion of macroscopic quantum superposition being unproblematic:

This is taking a ‘pragmatic’ stance  that does not really address the ontological issues.

-- Roger Penrose,  The Road to Reality (2004), p. 812

And, full-bore:

Many contemporary thinkers seem to have supposed that, in discarding its mechanist ontology, physics had discarded its ontology:  matter had been dematerialized … The very progress of physics itself  seemed to them to call for the renunciation of mechanism and materialism  in favour of the de-ontologised view of science presented by Mach.

-- John Watkins, Science and Skepticism (1984), p. 138

 

In philosophy proper, ontology is fundamental, being prior to anything else.   In its application to or rather analogue within  physics, by contrast, it historically comes behindhand, as a setting in order of what-all several centuries of reflection and experiment have come up with:  We may think of it as a kind of cast of characters, not fully drawn-up until the play has been written:  in the course of writing it, you find out you need a ladies-maid, and so eventually she is placed upon the prefatory page of Dramatis personae, that typographically precedes the play itself -- and as a nice afterthought, you name her Lisette.  Similarly, particle physics did not begin by being defined, a priori, as (back among the Greeks) the Science of Atoms, or (later) as the Science of the Proton, the Neutron, and the Electron, or (later still-- the Barock Age) as the Menagerie-management of the Particle-zoo (with a fixed given roster of inmates), nor as the Curating of the Wiggling of Strings.  There is a thematic continuity throughout all these stages, but the staffage keeps changing.

As for the role of this Ontology, or Cast of Characters, it is not (despite the spectral example of traditional metaphysics per se) just something to admire from afar, like Mount Rushmore, but rather, as Goedel said pragmatically re which axioms we should adopt for math and logic, they should themselves possess generative potential -- by their fruits ye shall know them.  Thus, hard-headedly:

Kepler’s theoretical ontology, unlike Gilbert’s, was not organically related to his laws;  even if it could be squared with the latter, which seems doubtful, it failed to make any contribution to the testable content of his system.

-- John Watkins, Science and Skepticism (1984), p. 197

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As remarked earlier, I may well go to my grave without ever grasping the concept of an observable, much less physical ontology in general.   Still, it is helpful towards organizing my thoughts, to have an online scribble-space, so that the matter is, so to speak, officially a topic, a project under way.   For now, this is just a whiteboard on which to stow some juicy quotes.  Your own juicy contributions are more than welcome.
For a more general surview of the ontology of the various sciences, click here.

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Physics may be defined as the art of saying things about stuff (or stuff about things -- predications concerning entities, for the fastidious).  But what are these entities, whereof we predicate?  In the first place -- observables.

P.A.M. Dirac, The Principles of Quantum Mechanics (1930; 4^th edn. 1958), p. 116:

From our assumption that the energy is an observable, there are sufficient stationary states for an arbitrary state to be dependent on them.

For a layman, this is bemusing.  The assumption that it’s an observable?   Can you observe it, or can’t you?  -- Evidently there is much more to qualifying as  “an observable” than merely being … observable.
(Compare Einstein, in one of his Zen moments: "It is the theory that decides what we can observe.")

P.A.M. Dirac, The Principles of Quantum Mechanics (4^th edn. 1958), p. 458 (re certain eigenstates):

Science contains many examples of theoretical concepts which are limits of things met with in practice  and are useful for the precise formulation of laws of nature, although they are not realizable experimentally, and this is just one more of them.

Emphasis added.  “Limits” in the mathematical sense.

Note that “not realizable experimentally” does not constitute much of a disability.  What, after all, is?  “Carthage lost the Punic Wars”; “I love you”; “E8 is a 248-dimensional rotation-space”:  no, almost nothing is.

Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p.402:

Quantities such as position, energy, momentum, and the like, capable of measurement… will be called observables,  although it is not intended to imply that they are observable directly.

The caveat is troubling enough;  but now this:

In quantum mechanics, the state of a system is no longer defined by means of a number of variables having an immediate intuitive appeal … In fact, it is not defined in terms of observables at all;  it is simply a function in configuration space.

Carl Hempel, “Problems and Changes in the Empiricist Criterion of Meaning” (1950):

Green, soft, liquid, longer than  designate observable characteristics, while bivalent, radioactive, better electric conductor, and introvert do not.

This odd assertion, by a well-known philosopher of science, seems more psychological than scientific.  It is reminiscent of Locke’s distinction between simple and composite ideas.

Eugen Merzbacher, Quantum Mechanics (1961, 2^nd edn. 1970), p. 153:

Following Dirac, we call observable any Hermitian operator which possesses a complete set of eigenfunctions.

This might sound opaque to some, but for a math guy it’s the clearest statement yet, by far.  Of course, what it amounts to physically, intuitively, is something else…

Gerald Holton, The scientific imagination (1978), p. 202:

The idea of making quantitative indicators of anything at all  fascinates some persons, and repels others as dangerous or absurd.  This difference is caused largely by thematically incompatible -- and therefore often unresolvable -- personal views concerning the ability of quantifiables to lead to … the deepest reality.

Note the silly dichotomy -- as though failing to lead to "the deepest reality" (a deeply suspect term) meant that they couldn't be "indicators of anything at all".

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I had some fun above, playing with a rumpled old word like stuff, shoving it before the microphone of science.  Here a gifted popularizer  makes similar play  with pronouns:

[In its] Einsteinian reframing … is spacetime a something?

-- Brian Greene, The Fabric of the Cosmos (2004), p. 39

In that historical context, the question concerned the ontological status of (the novelty) ‘spacetime’, as opposed to the traditional notions of the independent entities, space, and time.

(More recently, spacetime has been demoted in some theories -- not returning to a Cartesian product of space and time, but being derived as an epiphenomenon of more fundamental items.  Thus, twistor theory, among others.)

If there is no aether to provide the standard of rest, what is the what  with respect to which this speed is to be interpreted?

-- Brian Greene, The Fabric of the Cosmos (2004), p. 45

If an individual electron is also a wave, what is it that is waving?

-- Brian Greene, The Fabric of the Cosmos (2004), p. 88

(Here the wordplay inheres not in the pronoun what, but in the verb.  He could more conventionally have written, “What is the medium for the wave?”, but the startling verbal formulation ‘makes it strange’, confronting us with something more fundamental.)

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Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 75:

The fact that confinement prevents one from observing an isolated quark or gluon  might seem to make the whole notion of quarks and gluons as particles   somewhat metaphysical.  However, there is another property of the strong nuclear force, called asymptotic freedom.  The concept of these entities  was already well-defined, or not, as the case may be:  certainly well-defined as bookkeeping conventions, if nothing more.   Asymptotic freedom -- “at high energies, the strong force becomes much weaker, and the quarks and gluons behave almost like free particles” -- simply adds a further mode of observing their effects:  and in this case, their effects when they are relatively ineffectual -- quarks on holiday.

Failure to be observable in isolation certainly doesn't make a thing "metaphysical" (in the colloquial bad sense intended here).  You cannot observe a "brother" in isolation:  dissect him down to his last tissues, nothing will reveal his brotherhood but the historical context.  Nor, perhaps, can you observe Coulomb attraction in a single isolated particle -- it takes two to tangle.  (I might be wrong on this -- the photon cloud and all that.  But how does the cloud tell you whether you've got an attraction or a repulsion?)

Steven Weinberg, Dreams of a Final Theory (1992),  p. 181:

The positivist concentration on observables like particle positions and momenta  has stood in the way of a “realist” interpretation of quantum mechanics, in which the wave function is the representation of physical reality.

Wiki, "Quantum field theory" (excellent article, btw):

In quantum field theory, unlike in quantum mechanics, position is not an observable.

From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field.  Thus, the question ‘Why are all electrons identical?” arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.

The global phase of the wave function  is arbitrary, and does not represent something physical.

Wiki, "Implicate and explicate order" (of interest only to those who are already devotees of guru-physicist David Bohm):

 Bohm’s paradigm is inherently antithetical to reductionism … and can be regarded as a form of ontological holism.

Wiki, “Introduction to Gauge Theory”:

The electric field and the magnetic field are observable, while the more fundamental electromagnetic potentials V and A  are not.

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In this ontological context, it is far from clear how the phrase ‘more like’ is to be applied.  Comparison of historical theories gives no sense that their ontologies are approaching a limit:  in some fundamental ways, Einstein’s general relativity resembles Aristotle’s physics more than Newton’s.

-- Thomas Kuhn, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 265

Cf. too Dirac’s remarks (1951) that the aether concept was ripe for resuscitation.

[Update 8 May 2012] And now this:

http://www.nature.com/news/a-boost-for-quantum-reality-1.10602

The philosophical status of the wavefunction — the entity that determines the probability of different outcomes of measurements on quantum-mechanical particles — would seem to be an unlikely subject for emotional debate. Yet online discussion of a paper claiming to show mathematically that the wavefunction is real has ranged from ardently star-struck to downright vitriolic since the article was first released as a preprint in November 2011.

The paper, thought by some to be one of the most important in quantum foundations in decades, was finally published last week in Nature Physics

They say that the mathematics leaves no doubt that the wavefunction is not just a statistical tool, but rather, a real, objective state of a quantum system.

I told you so...

Physicists reify space-time. They elevate it from a four-dimensional diagram used to record their experience into the kind of “real essence” that Bohr warned us not to seek.
-- David Mermin (March 2014), at:

http://www.nature.com/news/physics-qbism-puts-the-scientist-back-into-science-1.14912?WT.ec_id=NATURE-20140327

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Not the same as the question of the building-blocks (ontological bricks) of physics, but related to it, is that of the Boundaries of Disciplines:  between physics and neighboring fields (chemistry, mathematics, …) and within physics itself (mechanics, astronomy, electromagnetism, condensed-matter, nucleonics, quantum theory, …).   In one sense, the question is idle -- you are working on whatever project you are working on, with methods appropriate thereto, however outsiders might classify them.  But it also has practical consequences, e.g. in the writing of textbooks.  As:

The traditional teaching of thermodynamics and statistical mechanics  as distinct subjects,  has often left students with their knowledge  compartmentalized, and has left them ill-prepared to accept newer ideas such as spin temperature or negative temperature  as legitimate and natural.

-- F. Reif, Fundamentals of statistical and thermal physics (1965), p. viii

That, from the textbook we used in stat mech at Harvard -- in the physics department, though previously I had only met notions of enthalpy, temperature, free energy, and entropy, in a chemistry course.

Similarly, Lindsay & Margenau remark, in their historical overview Foundations of Physics (1936), that they are moving away from treating optics and electrodynamics as distinct disciplines, “the former being, since Mawell’s time, really a branch of the latter.”

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God’s-truth vs Hocus-pocus:

It is tempting to dismiss these quantum waves  as mathematical contrivances … but in the laboratory these “probability waves” can be manipulated with mirrors …

-- George Johnson,  A Shortcut Through Time (2003), p. 38

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The prototypical example of an ontological ‘bit’ of chemistry and physics, is the atom (the ‘indivisible’ in its Greek etymology).  But later perspectives can get quite unprototypical:

A neutron star … is basically a giant atomic nucleus, stabilized by gravity.

-- J. Richard Gott, The Cosmic Web (2016), p. 29

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Esse est percipi (redivivus ter)

“To be is to be perceived” -- Bishop Berkeley
[corresponding to the Latin   esse est percipi, pronounced ESS-ay est pare-KIP-ee.  Percipi is the passive infinitive, a delightful category, which does not exist in any of the other languages with which I am conversant.]

“We call a real dynamical variable whose eigenstates form a complete set   an observable.”  -- Paul Dirac

“To be  is to be the value of a variable.”  --  W.V.O. Quine

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~

~  Posthumous Endorsement ~

"If I were alive today, and in the mood for a mystery,

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am Bishop Berkeley, and I approved this message.)

~         ~

~

Three radically different takes on  being and seeing.

(Cf. "Seeing is believing" -- vs. Berkeley:  Being seen  is being.)

The first and most famous of the epigrams above  applies to the ontology of basic objects, such as coffee-cups -- about which no plain man has any doubts whatsoever.   (Quine's quip  will, however, frequently detain us further;  e.g. here.)

A related but different question  concerns things which, unlike a coffee-cup or its relativistic mass, you cannot directly perceive -- though someone else might.

Scott Soames, Philosophical Analysis in the Twentieth Century (2003), vol. I, p. 17, re the views of G.E. Moore:

For things presented in space, but not things to be met with in space, to exist is to be perceived.  That is, afterimages .. and pains  can only exist when they are perceived or experienced.

Or, classically,  “Beauty is in the eye of the beholder”.

A psychological vice ontological version of Bekeley’s epigram, is the vernacular “Out of sight, out of mind.”  (Cf. French, "Si tu te tais, ni vu ni connu.")  Or, as Fielding wittily put it in Tom Jones,

De non apparentibus, et non existentibus,

eadem est ratio. -- In English, ‘When a woman is not seen to blush, she doth not blush at all.’ 

(Lawfolk interpret this Latin tag more prosaically:  “What is not juridically presented cannot be judicially decided.”   Spoilsports.)

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A noted intuitionist philosopher, anent “the celebrated [loathsome, leprotic -- ed.] thesis that mathematical statements do not relate to an objective mathematical reality  existing independently of us”, writes that, on such a view, to be is to be conceived:

Unlike material objects,  mathematical objects are, on this thesis, creations of the human mind.  They are objects of thought, not merely in the sense that they can be thought about, but in the sense that their being is to be thought of [or rather, thought into existence -- ed.];  for them, esse est concipi.

-- Michael Dummett,  “ The Philosophical Basis of Intuitionistic Logic”, in: Truth and other enigmas (1978), p.  228

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The above is really just a placeholder post, reminding me to someday address the deeper problem of observables.   This may never happen.  To begin with, even in the simplest quantum-mechanical case, an observable is an operator on Hilbert space, so you have to wrap your mind around that.  With some luck and some study, that might be doable.  But just now I encountered the following dismaying sentence:

In quantum field theory, unlike in quantum mechanics, position is not an observable.

I may well have retired to my hamster farm, before ever figuring such things out.

In the meantime, some further thoughts about ontology  here.

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Bonus quote:

  How do you know you’re having fun  if there’s no one watching you have it?

  --Douglas Adams, The Restaurant at the end of the Universe (1980).

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Visitors to the site have enquired about the pronunciation of the classic Latin phrase “esse est percipi”.   About this (although I used to be editor of pronunciation at Merriam-Webster, back in the day) I have so far had nothing to say.  Embarrassed, I messaged my colleague and comrade Dr. Keith Massey (officially out of comms, but reachable in the bush), as follows:

More than once, someone has searched the blog via
   pronounce esse est percipi
-- alas in vain.  In my head, I have always pronounced this pear-KIP-ee, but, given that the vowel is short, the Latin would be more like PEAR-kip-ee, no?
Or rather, the Vulgar Latin, since Latin itself, being quantitative, lacked phonemic lexical stress.  Correct so far?
A second question is how philosophers in various countries pronounce it -- anglicized, frenchified, etc.

Swifter than lightning, our colleague replied:

In classical Latin the C is always pronounced as a K. But we don't know when the thing started to change before E and I. I suspect it was already being realized regionally as something else in the Late Imperial period. And speakers of Latin dialects took to pronouncing their classical Latin as if it were their living register. And so, in France,  percipi was pronounced “persipee”. In Italy perchipee, and in Spanish pershipee (attested still in Ladino) and later, in Iberia as perthipee, and, yet later in the New World, as persipee.

Accent in Latin is a hotly debated topic. I personally follow the view that the classical accent is unknowable and therefore all we have to work from is the living dialects and the rules of Ecclesiastical Latin. And so, penultimate except in a few cases. Accent falls in the syllable before -ibus. But I'd say perCIPee.

As for Anglo-Latin, this is an exciting topic. There's some evidence that a Romance language survived in the Isles for a several centuries after the withdrawal of forces in 380. It apparently, from inscriptional evidence, turned long E into a long I, fecit --> feecit. The way we pronounce phrases like Habeas Corpus may actually be representative of the pronounciation of Anglo-Romance, rather than a later scholastic recasting.

Those further interested in this topic can consult the essay by Thomas Pyles, “The Pronunciation of Latin in English:  A Lexicographical Dilemma”, reprinted in his Selected essays on English usage (1979).  Executive summary:  The history is so tangled, it is now an unresolvable mess.

In his “Tempest in Teapot: Reform in Latin Pronunciation”, reprinted in the same volume, Pyles quotes an amusing epigram that alludes to what became of Latin in Spain, where original v and b merged into a bilabial fricative:

~ Felix natio, ubi vivere est bibere ~

[Variant: "o felix iberia, ubi vivere est bibere"]

For further adventures in pronunciation, click here. 

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Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

http://www.linguasacrapublishing.com/justice.html

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Distinguishing discontinuous state-reduction (which he dubs R) from linear evolution as per the Schrödinger equation -- i.e., the “collapse of the wave-function” upon “observation”, a mathematician remarks:

I do not mean to imply that the experimenter deliberately sets up a ‘measurement’ to achieve this.  … Nature herself is continually enacting R-process effects,  without any deliberate intentions on the part of an experimenter or any intervention by a ‘conscious observer’.

-- Roger Penrose,  The Road to Reality (2004), p. 593

Thus achieving the esse of percipi  ‘naturally’ (vacuously).

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On Vulgar Numbers

[Note:  The title is a pun.  Cf. vulgar fractions, i.e. common fractions (as opposed to continued fractions, etc.)]

Among Harvard math-majors, back in my day (before your time -- don’t ask), the commitment to abstraction was intense and unquestioned.   We had barely begun to shave -- and many of us had never been laid -- but we took to abstraction like -- like a duck to water, like a kitteh to cheezeburger.   It was to some extent an end in itself:  not the pitch of wisdom by any means, but temporarily no more harmful than any other young-man’s idealistic infatuation, the blaue Blume or what-have-you.  No sooner did we learn about a thing, than we wanted to generalize it.  Actual numbers had been replaced by x and y in junior-high algebra;  now we rushed to embrace the abstract structures in which these variables lived:  groups, rings, fields … The brighter among us (I was not of their number) before graduation  came to embrace  what its own practitioners called “abstract nonsense” (category theory, “diagram-chasing”), the night or perhaps twilight  in which all structures are grey.  As for the everyday numbers of science -- ungainly things with decimal points in them -- they were as attractive as a turd on a sidewalk.

Accordingly, we had no truck at all with what we called “apple-math” (punning on Appl. Mathematics, the course-catelogue designator, plus the notion of counting up fruit).  Such a department did exist, so we were told, probably somewhere out behind the barn;  but our steps would never take us there.   Oddly, the College itself shared our prejudice, it would seem, since Appl.-Math majors were denied the B.A. shared by math majors and literature majors alike:  they got a B.S., along with (presumably) majors in Sports Medicine  or Veterinary Science.   It told the select world:  “These fellows know how to ply a slide-rule, but they have not received the education of a gentleman.”

[For a glimpse of freshman calculus at Harvard, back when the world was young,
click here:
http://worldofdrjustice.blogspot.com/2011/12/adventures-in-algebraic-geometry.html

And for sophomore year -- the legendary "Math 55", fabled in song and story:
http://worldofdrjustice.blogspot.com/2013/03/andrew-gleason-in-memoriam.html ]

Note:  Our Harvard freshman prejudice  has been shared by others.  A great Chinese mathematician reminisces:

Arriving in California at the age of twenty… I had no idea of what direction to pursue.  I was initially inclined toward operator algebra, one of the more abstract areas of algebra, owing to my vague sense that  the more abstract a theory was, the better.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 35

Appendix for physicists:

The most practical person must realise that abstract arguments (by which we really mean  arguments with a tremendously wide range of applicability  are a necessity … now that science has grown so vast.  If the engineer is willing to overcome his nostalgia for the practical, and embark on the study of Lagrange’s equations in a apirit of abstraction, he will be rewarded by having at his disposal a powerful tool for the study of electrical networks, which are not ‘dynamical systems’ in the ordinary mechanical sense, but nonetheless behave as if they were.

-- John Synge & Byron Griffith,  Principles of Mechanics (1942, 1959), p. 411

Graduate students in theoretical physics … are very often impressed with “formalism” -- the formal apparatus of their subject. … I suffered … from an infatuation with beautiful formalism.  Working with Viki Weisskopf was a most effective remedy against the excesses of such an infatuation.  He never ceased to harp on the importance of … understanding, by means of simple arguments, the physical meaning of a theory …

-- Murray Gell-Mann, “The Garden of Live Flowers”, in Selected Papers (2010), p. 27

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Here, now, is a very funny reflection of a variety of attitudes about our number-friends:

http://xkcd.com/899/

Explanatory footnotes here:

http://www.explainxkcd.com/2011/05/16/number-line/

Re the locus “2.9299372”, the explainer acutely comments:

2.9299372 is a President's Day reference because it is the average of e and pi just as the American President's Day is always observed on a random day between George Washington and Abraham Lincoln's birthdays.

There is, however, another layer to this.   The notion of “observed value” is strictly from experimental science, especially physics and chemistry.  Thus, the “observed mass of the proton”.   In that sense, you could speak of an “observed” value of pi, meaning the length of a tape-measure wrapped around a circle of diameter one.  So here an image is conjured up, of e and pi being within experimental error of each other.   If you just consider them as regular old numbers, like the mass of the proton and the mass of the neutron, that even makes sense -- presumably there was a time when those two masses were not neatly differentiated.    But to a pure mathematician, the notion is riotous -- risible:  e and pi each thrones separately and uniquely in a starry empire of which they are sovereign.  Their structural status is everything;  their particular numerical value -- nothing.

There is, indeed, a big difference in ontological status between “observed quantities” and the pure numbers of mathematics.   As, in the case of mass, we can’t judge well visually ourselves, so we use a scales (or a cloud-chamber) and let that “observe” it for us.  In the case of color, sometimes the scientific intstrument is just our eyes.  So, “3.8 kg.” and “vermilion” have similar ontological status.  Whereas a number (in particular an integer, or pi), in pure mathematics, has no observational status at all.  Indeed, if anything, pi observes you -- it’s liable to pop out at you without warning.  You’ll be sitting there innocently summing up an infinite series (say of the reciprocal of squared integers), and it turns out to be pi-squared over six.

Commenting on “if you encounter a number higher than this, you’re not doing real math”, the explainer gets things backwards, I think.  It is not that discrete-math practitioners don’t think that the rest is not real math;  it’s that mathematicians in general, of the abstractophilic stripe, have little to do with big numbers in general, because these do not (like pi, or 2) constantly pop up when you’re not looking for them.   True, number theorists consider indefinitely large integers:  but collectively, with little interest in this one or that one in particular.   Thus, the Goldbach conjecture applies to all integers at once.  Should someone ever find a smallest counterexample, then that integer will be enshrined, for a time; if there are infinitely many counterexamples, its significance will shrivel.  Intense interest in a menagerie of specific large integers tends to be best represented, I suspect, among idiot-savants.

There are, however, a couple of comparatively recent developments  that really do care about -- or at least come across -- specific large integers:  namely, the (successful) classification of all finite simple groups, and  investigations of the exceptional simple Lie groups.

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Finally, a rich index to a plethora of math-related posts:
http://worldofdrjustice.blogspot.com/2014/08/miscellanea-mathematica-ter-renovata.html